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INTRODUCTION: THE THIRTEENTH ROOTS


Although 13th roots are the official tasks as integer roots, for the mental calculation world records and with objective reasons, many people do not know the geniality of 13th roots.
Most material in this site is about integer roots.
The easiest example is the 13th root of 8192, which is 2.
8192 is 2 raised to the 13th power.
In this page we explain mathematically and numerically, regardless of the world records, why we choose the 13th roots instead of cube roots, fifth roots, 9th roots or 10th roots.



There are roots which are easier than the 13th root, such as the 11th root or the 667th root, and roots which are more difficult

If 2 roots get the same difficulty, the real difficulty lies in the number of possibilities of the problem:
the 13th root of a 100-digit number (8 millions of possibilities)is a more difficult task than the 137th root of a 1000-digit number(330 000 possibilities)

Order of difficulty for the different roots(examples:)

10>15>12>13=17=137=23=7>667>9=19>11=31>101>1001>4>3>2>1



Firstly, 13 is a prime number. In the case of 9th roots, we can calculate the cube root, then the cube root. This is not possible for 13th roots.

In addition, 13 is the first 2-digit number which has a solution n=3 for the equation 4n+1=13.
It belongs to this sequence: 1,5,9,13,17,21,25... These numbers and their successors are all the cases where the last digit of the root is always the last digit of the power:


013= 0
113= 1
213= 8 192
313= 1 594 323
413= 67 108 864
513= 1 220 703 125
613= 13 060 694 016
713= 96 889 010 407
813= 549 755 813 888
913= 2 541 865 828 329
...

Thus we find the first elementary rule:
To calculate the last digit of the 13th root you just have to copy the last digit of the 13th power.

But why not the 5th root, the 17th root or 21st root?
We have to point out that in a lot a cases the 21st roots keep the last 2 digits(13th roots keep the last digit), the 101st keep the last 3 digits and the 1001st root keep the last 4 digits. However, these extended properties show always some exceptions.


More interesting properties for the 13th roots can be found by looking at the array below;
It gives the 2-digit endings of the 13th powers of the 2-digit endings, not divisible by 10[in the case of ending 0, write 0 and forget the 13 trailing 0 of the power to continue the process].


-1 -2 -3 -4 -5-6-7-8-9
0- 01 92 23 64 2516078829
1- 31 72 53 44 7596376859
2- 61 52 83 24 2576674889
3- 91 32 13 04 7556972819
4- 21 12 43 84 2536270849
5- 51 92 73 64 7516578879
6- 81 72 03 44 2596876809
7- 11 52 33 24 7576174839
8- 41 32 63 04 2556472869
9- 71 12 93 84 7536770899

We have to remark several things:

Thus a caracteristic of the 13th roots comes:
A bijective category exists, where as roots ending in 5 like 5th root, 25th root,... do not have a bijective category. Only root orders ending in 9, 13, 17 and 21 ,+20k for each order,have a bijecive category.

But this is not all!
Look at the bijective category.

Central antisymetry and arithmetic sequence lead to this minimal array for the bijective category:

-1-3
0123

The equivalence 13th root/77th power



Because the 1001st power (1001=13x77) keeps the last 4 digits, we get in the case of root endings 1,3,7,9 the following property:

X13=Y<=>X=Y77 modulo 10000





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